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The isoperimetric conjecture for the cubic flat three-torus (balls, tubes, slabs)

In the unit cubic flat torus R3/Z3\mathbb R^3/\mathbb Z^3, which regions of prescribed volume VV have least perimeter? The conjecture predicts that the minimizers are, in turn, round balls, solid circular tubes about shortest closed geodesics, and slabs between parallel coordinate tori, followed by their complements, so the profile is min⁡{(36π)1/3v2/3,2πv,2}\min\{(36\pi)^{1/3}v^{2/3},2\sqrt{\pi v},2\} with v=min⁡(V,1−V)v=\min(V,1-V). Small volumes (Morgan-Johnson) and volumes near one half (Acerbi-Fusco-Morini) were known, and Milman reduced the profile to the two transition volumes, but excluding higher-genus competitors in between remained open. Is the ball-tube-slab picture the true isoperimetric profile of the cubic three-torus?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Geometric measure theory; periodic isoperimetric problem
Posed by
Hauswirth, Perez, Romon and Ros, The periodic isoperimetric problem, Trans. AMS 356 (2004); also A. Ros, The isoperimetric problem (Clay Math. Proc. 2, 2005), Section 1.6
Year posed
2004
Years open
22y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
28 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims the full isoperimetric profile of R3/Z3\mathbb R^3/\mathbb Z^3, I(V)=min⁡{(36π)1/3v2/3,2πv,2}I(V)=\min\{(36\pi)^{1/3}v^{2/3},2\sqrt{\pi v},2\}, and a classification of all finite-perimeter minimizers: balls for V≤4π/81V\le4\pi/81, tubes about shortest geodesics for 4π/81≤V≤1/π4\pi/81\le V\le1/\pi, coordinate slabs for 1/π≤V≤1/21/\pi\le V\le1/2, complements above 1/21/2, with exactly the two adjacent types at each transition volume. The new step excludes three- and four-vertex nonstandard configurations at the two transition volumes via nested planar sections. It does NOT treat non-cubic flat tori, the cube with relative perimeter beyond the reflection correspondence, or higher dimensions.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. Some one-variable estimates are closed with explicit outward-rounded arithmetic bounds in an appendix. The full profile and classification have a Lean comparator challenge in the release.

Verification

No independent mathematician has checked this yet. The main theorem was read against the HPRR and Ros formulation. The Lean challenge lean/ComparatorChallenges/CubicTorus.json (solution module OAI.Geometry.CubicTorus.Main, present at the pinned commit) is not in the formalization catalogue; it was found through lean/docs/354.md. Its statement OAI.CubicTorus.unit_cubic_isoperimetric was read here: for every 0<V<10<V<1, with De Giorgi perimeter defined by smooth periodic test fields, a minimizer exists, every minimizer has the candidate-profile perimeter, and the minimizers are exactly, up to a torus isometry and a null set, the canonical ball, tube or slab on closed volume ranges, or complements for V>1/2V>1/2. This is the headline. Not rebuilt here.

Sources

Changelog1 change

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